Characterizing classical minimal surfaces via the entropy differential

TL;DR

This paper introduces the entropy differential on minimal surfaces in Euclidean 3-space, characterizes classical examples, and establishes a curvature estimate and compactness theorem based on this new invariant.

Contribution

It defines a new meromorphic quadratic differential called the entropy differential, linking it to geometric conservation laws and characterizing classical minimal surfaces.

Findings

Characterization of classical minimal surfaces via the entropy differential

A new curvature estimate for embedded minimal surfaces with small entropy differential

A compactness theorem related to the entropy differential

Abstract

We introduce on any smooth oriented minimal surface in Euclidean $3$-space a meromorphic quadratic differential, $P$, which we call the entropy differential. This differential arises naturally in a number of different contexts. Of particular interest is the realization of its real part as a conservation law for a natural geometric functional -- which is, essentially, the entropy of the Gauss curvature. We characterize several classical surfaces -- including Enneper's surface, the catenoid and the helicoid -- in terms of $P$. As an application, we prove a novel curvature estimate for embedded minimal surfaces with small entropy differential and an associated compactness theorem.